# Variant of the Maximum-Modulus Principle

## Homework Statement

Show that the maximum principle holds with $$\phi :\mathbb{C}\rightarrow\mathbb{R}$$,

$$\phi(z) = (\textrm{Re}(z))^4 + (\textrm{Im}(z))^4$$,

in place of the modulus:

If $$U \subset \mathbb{C}$$ is open and connected, $$f : U \rightarrow \mathbb{C}$$ is holomorphic and $$p \in U$$ is such that

$$\phi(f(p)) \geq \phi(f(z))$$

then $$f$$ is constant.

Can one also replace the modulus by the sine of the modulus, i.e. the function with $$\phi(z) = sin(|z|)$$?

## Homework Equations

If $$z=x+iy$$

$$\phi(z)=x^4+y^4$$

## The Attempt at a Solution

I tried using the open mapping theorem and the fact that

$$f(U)\subseteq\{z|\phi(z)\leq\phi(f(p))\}=\{z|x^4+y^4\leq M\}$$

where $$M=\phi(f(p))$$, but was not successful.